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Boussinesq Bubbles
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Investigation of singularities in Boussinesq convection
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<Center> Michael Minion </Center>
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<!WA0><IMG SRC="http://math.nyu.edu/postdocs/minion/res/color_bubbles.gif" ALT="COLOR BUBBLE">
<P> 
The figure above was produced using a Godunov/Projection method
applied to the two-dimensional equations for Boussinesq convection.
The initial conditions for this computation were taken from
a paper by E and Shu and consist of a smooth "bubble" of density
in a zero flow field.  As the bubble begins to rise, strong
fronts in the density field form (shown by the contour lines in the
figure) and vorticity is baroclinically produced along this front
(shown by the color shading).
As the numerical method begins to under-resolve the flow, the
density front destabilizes and rolls up under the influence
of the vorticity.  Because of the experience gained from
studying the appearance of <!WA1><a href="http://math.nyu.edu/postdocs/minion/res/under.html">spurious vortices</a>
 in under-resolved flows
the correctness of the front roll-up is questionable despite
the fact that it is consistent with the evolution of the bubble.

In order to investigate this problem further I have developed
adaptive mesh refinement techniques for the Boussinesq equations.
Below is an example of a simulation using initial conditions from
the paper of Siggia and Pumir.
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<!WA2><IMG SRC="http://math.nyu.edu/postdocs/minion/res/s3ac.gif" ALT="REFINED BUBBLE">
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<H3> Impact: </H3>
Besides providing an excellent test problem for the development,
improvement and evaluation of adaptive mesh refinement methods
for incompressible flow, I hope to also shed some light on
the open and sometimes controversal question as to whether or
not the two-dimensional Boussinesq equations admit finite
time singularities.
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<H2> References </H2>
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<LI>
   Weinan E and Chi-Wang Shu,	
   <B>Small-scale structures in Boussinesq convection</B>,  
    <CITE> Physics of Fluids, </CITE>	January 1994.
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   Alain Pumir and Eric D, Siggia,	
   <B>Development of singular solutions to the axisymmetric Euler equations</B>
    <CITE> Physics of Fluids, </CITE>July 1992.
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A paper containing results from my study is in preparation.
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<!WA3><A HREF="http://math.nyu.edu/postdocs/minion/index.html"> 
To Michael Minion's Homepage</A>
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